Crank J. Free and Moving Boundary Problems
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422
Subject index
seepage faces, hodograph mapping
of
290
seepage phenomena, examples
of
30,
31
seepage problems, analytical solution
of 283-312
seepage surfaces, defined 32
semi-inverse methods 22-3
semi-permeable membranes, fluid transfer
across 8
separation point, disc-in-cylinder 326, 332-
5
series solutions
melting ice 125-6
slab-melting problem
131
shallow-flow approximation method 288
sheetpile, rectangular dam
44-6,321,382
shocks, hyperbolic equations involving
217
shooting techniques
method
of
lines
182
rectangular dam with capiJIary fringe 78
short-time solutions
melting/solidifying slabs 125--6, 128
oxygen diffusion problem
120--1
silty soil problem 279
similarity solutions
cylindrical coordinate
111
defined
101
density change 10
moving heat source 111-14
multi-phase 106, 107, 108
Neumann's 101-10
spherical coordinate
110--11
supercooled liquid 104, 105, 108
similarity variables
cylinder solidification 162
sphere solidification
162
simple dam problem
Alt's numerical solution 382
Baiocchi transformation 34-8, 99
boundary singularities 332-6
Cryer algorithm
360--2
Dupuit formulae 284-5
fixed domain method 35-8
hodograph mapping
of
291,297
minimization formulation
of
361-2
numerical solution 313-16
physical derivation 31-4
Polubarinova-Kochina solution 300--12
stream functions 34
extended to fixed domain
4{}-1
variable-interchange method 344-7
variational inequality formulation 38-40
Simpson formula 120, 176, 392
simultaneous diffusion and heat flow,
14-
16
single-dimension
see
one-dimension
single-phase problems
enthalpy method for 219
fixed grid finite-difference method
for
165--6
heat-balance integral method used
in 129, 132,
133
steady-state approximation for
14{}
single-phase Stefan problem 2-4, 11,
99-
100,114
boundary-fixing transformation for 187
finite-element method for 173-81
isotherm migration method for 199-201
spatial subdivision in 142-3
variational inequalities for 269-75
single-phase two-dimensional problem
body-fitted coordinates used in 195-7
singularities
boundary 332-6
finite-difference grid spacings near 315
modified grid finite-element methods
dealing with 177-8
separation point 326, 332-5
special finite elements near 316
sloping-base dams
50--2
Boussinesq equations for 286
slanting inlet face 51-2
smoothed enthalpy functions 227,229-30
smoothed enthalpy method, boundary-
position determined by 233-4
solid mechanics, examples 1
solid slab problem, Green's functions used
in
117
solid-liquid zone models 6
solidification
constant-rate 2, 144-7
volume change
on
103
solidification front, moving heat source
treatment 111-14
space vehicle, ablation
of
alloy walls
of
12
spatial subdivision
heat-balance method 133-7
steady-state approximation 142-4
sphere problem
constant-rate solidification
145--6
solutions
of
156-62
spherical coordinates, similarity solutions
in
110--11
split-domain method
arbitrary-shaped dam
54-7,373-6
open channel 66
pond seepage 70
spot welding examples 5, 240
square cylinder, melting
of
275
square duct, melting around
228
square region, solidification
of
199, 279
square-channel freezing 236
stability analysis, melting/solidifica-
tion
150--1
starting solutions
isotherm migration
method
200
melting slab 126
steady state
approximations using 139-42
axisymmetric wells 66-7
oxygen diffusion
19,20
Stefan condition
4-5
IMM form
of
214
supercooled liquid 104, 108
Stefan jump condition 221, 231
Stefan number 7, 9,
198
Stefan problem
asymptotic solution 156
concentrated thermal capacities in 23-9
convective boundary condition 141
defined 1-2
finite-element methods applied
to
173-8
general solution 2
generalizations 5-29,
281
homogenization
of
8
implicit boundary conditions 19-23
inverse 11-12, 144-7
laser hole-cutting 152
moving heat-source description 229
multi-phase 12-14
non-linear
148
one-dimensional 117, 173-8
simple example
2-5
single-phase
see
single-phase Stefan
solidification
of
sphere 156-62
three-dimensional 16-19
three-phase 178
two-dimensional
16-19,274-5
two-phase
4,11,222,227,236-7,275-81
stellar evolution problem, modified bound-
ary conditions 332
stepwise time history, enthalpy
method 238
stiff-system algorithm 191-2
stratified dams 71-4
Alt's numerical solution for 383-4
boundary singularities
in
336
Cryer algorithm for 363
stratified media, Dupuit formulae applied
to
285
stream functions
complex notation
of
291
rectangular dam with toe drain
41-2
simple dam 34,
4{}-1,
291, 335-6
stream1ines 34, 61, 90
successive overrelaxation
(SOR)
algorithm
capillary-fringed dam 78
dam with inlet face slanted 49
enthalpy method 232-3
Hele-Shaw injection 85
method
of
lines 184-6
Subject index
423
modified block 356-9
open-channel seepage
65
oxygen diffusion problem 261-3
point 355-7, 359
sloping-base dam 52
also
see
Cryer algorithms
supercooled liquids, solidification
of
104-
9,111
superheated solids, melting
of
109
surface tension effects, Hele-Shaw flow
82
T-plane, hodograph mapping 295-7
Taylor series
oxygen diffusion problem 166-7
single-phase Stefan problem 116
Stefan problem 2,
116
variable space grid finite-difference
method
172
temperature distribution, successive
freezing 238-9
temperature profiles
sphere solidification
156
steady-state 140, 141-2
supercooled liquid
108
temperature-time histories, constant-rate
solidification 145-6
temperature-dependence
alloy solidification 15
enthalpy function 218
generalized Stefan problem 8
temperature-dependent thermal
properties 132, 187
temperatures
concentrated capacity 27-8
constant-rate solidification
2,
147
terminal temperature distributions
solidification
of
molten sphere
16{}-1
test functions
finite-element method 178, 180
generalized
dam problem 80
thermal capacities, concentrated
23-9
three-dimensional dam problems 57-60
coordinate interchange used in 353-5
three-dimensional problems
explicit fixed finite-difference grid
methods for 168
method
of
lines 183
variable-interchange
method
for 347-53
three-dimensional Stefan problems 16-19
three-eights rule 120
three-phase problems 12-13
finite-element method used
t{)
solve
178
time steps
adjustment
of
168,
201
variable 168-70
time-dependence
electrochemical machining 88-9
424
Subject index
time-dependence-continued
general-shaped dams 90, 92-3
Hele-Shaw injection 82-3
oxygen diffusion problem
120--1
Stefan problem
173-8,317
toe
drains 41-4, 46, 49,
324
transformations
Baiocchi
see
Baiocchi transformation
Boadway
2()4...{),
344,
348
boundary-fixing 187
conformal
293-7,298,299,300
coordinate 93, 187,
189
Duvaut
270,271,276
equipotential 197
Fourier 117, 187,263
Hamel
29~7
IMM variables
complete 204-11
partial 201-4
integral 114-17
inverse 115, 294-5
Kirchhoff 219,281
Kozeny
43
Laplace 114, 115, 153, 154, 158, 159
multi-variable interchange 353-5
Oberkampf 194, 196, 197
Ricatti 182,
186
Schwarz-Christoffel 293-5, 299
single variable interchange 343-53
Winslow 197
Zhukovski 295, 298, 300
transient problems
90--1,
321
trapezoidal dams, seepage through 312
trapezoidal drainage ditch, hodograph
method 297-300
trapezoidal rule
183
trial free-boundary methods 313-32
trial-and-error process, enthalpy
method 230
triangular channels, seepage from
63
triangular dam, hodograph planes for 291-
2
triangular finite elements 56, 174, 380
truncation methods 253-7
linear complementarity methods com-
pared with 267-9
oxygen diffusion 23, 255-7
two-dimensional annular electrochemical
machining problem 385-93
two-dimensional dams
see
dams
two-dimensional heat
flow
problem,
isotherm representation 212, 214
two-dimensional model, oxygen
diffusion 21, 257
two-dimensional problems
explicit fixed finite-difference grid
methods for 168
method
of
lines for
183
two-dimensional solidification front, heat-
balance integral method applied
to 137-9
two-dimensional Stefan problems 16-19
variational inequality formulation
of
274-5
two-dimensional two-phase problems
linear finite element solution 279
variational formulation 275
two-phase, multi-dimensional problem, en-
thalpy method for 218-19
two-phase melting-slab problem, heat-
balance method applied
to
130
two-phase moving boundary problem
16-
17
two-phase Neumann problem 101-2,
147
two-phase solidification problems, explicit
fixed finite-difference grid methods
168
two-phase Stefan problem 4,
11
nodal enthalpy 236-7
variational inequalities 275-81
weak solution for 222, 227
two-phase systems, dimensionless express-
ion
of
9
unsaturated
flow
75-9
unsteady porous
flow
with accretion
286
dam
90
variable space grids
170--3
variable time step methods 168-70
enthalpy method 236-7
Goodman integral method <:ompared
with
170
variable-interchange methods 336-55
adaptive variational formulation 337-43
multi-variable interchange 353-5
single variable interchange 343-53
variational inequalities 257-82
arbitrary-shaped dam, with
toe
drain
56
axisymmetric
flow
67, 68
capillary-fringed rectangular dam 77
coastal aquifer
61
dam with inlet face slanted 47-9
electrochemical machining
85,86,87
evaporation
75
general derivation
of
100
Hele-Shaw injection 84
journal bearing lubrication 96
mathematical properties
of
281-2
minimization formulation
of
265-6
rectangular dam
with sheetpile
45,46
with
toe
drain 42-3, 46
variational inequalities-continued
simple rectangular dam problems 38-40
single-phase Stefan problem 269-75
sloping-base dam 50
stratified dam
73
theoretical properties
of
257
three-dimensional dam
57
transient dam 92-3
two-phase Stefan problem 275-81
well problem
31,67,68
variational inequality methods
355
velocity potentials
boundary singularity 334-5
complex notation of
291
Hele-Shaw
flow
82
simple rectangular dam 32
vertically stratified dams 71-3,
336
volume change, solidification
103
water table, seepage into 63,
65
water/air interfaces, saturated
flow
30
Subject index
425
weak solutions
capillary-fringed rectangular dam
77
enthalpy method 217,220-3
stratified dam 72
well problem
67
wedge-shaped space freezing 113-14
welding problems, 5-6, 239-41
well problems
28-9,31,66-70,372-6
Dupuit formulae applied
to
285
numerical solution
of
372-6
variational inequality solution of 31,
372-6
wells
coastal 60-3
hot-liquid injection into
28
Winslow transformation
197
zero latent heat 147, 148, 150
zero specific heat 109, 242-5
zero-order perturbation solutions,
laser hole-cutting 153, 154
Zhukovski function 295, 298,
300